Continuous-time online distributed constrained optimization via unbalanced digraphs

Abstract

In this paper, online distributed constrained optimization is investigated by employing a continuous-time multi-agent systems. The objective of the agents is to cooperatively minimize the sum of time-varying cost functions subject to a convex set at each time. Each agent can only have access to its own cost function and the convex set, and cost function in the future is not available. To address this problem, we propose a modified online distributed "projection+gradient" algorithm, which involves each agent minimizing its own cost function while exchanging local state information with others via an unbalanced digraph. Performance of the algorithm is measured by dynamic regrets. Under mild assumptions on the graph, we prove that if the rate of a minimizer's variation is within a certain range, then regrets, as well as the violation of constraint, grow sublinearly. A simulation is presented to demonstrate the effectiveness of our theoretical results.